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Bifurcation analysis of an age-structured SIRI epidemic model

  • In this paper, an SIRI epidemic model with age of infection and the proliferation of susceptible individuals with logistic growth is investigated. By using the theory of integral semigroup and Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that if the threshold parameter is greater than unity, sufficient condition is derived for the occurrence of the Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.

    Citation: Xiaohong Tian, Rui Xu, Ning Bai, Jiazhe Lin. Bifurcation analysis of an age-structured SIRI epidemic model[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7130-7150. doi: 10.3934/mbe.2020366

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  • In this paper, an SIRI epidemic model with age of infection and the proliferation of susceptible individuals with logistic growth is investigated. By using the theory of integral semigroup and Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that if the threshold parameter is greater than unity, sufficient condition is derived for the occurrence of the Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.


    In the past decades, great attention has been paid by many researchers to SIR epidemic models [1,2,3,4,5,6,7], in which total host population is divided into three classes called susceptible (S), infective (I) and removed (R), and the immunity that is obtained upon recovery is assumed to be permanent. In [8], for herpes viral infections, considering the fact of recovered individuals may relapse with reactivation of latent infection and reenter the infective class, Tudor proposed the following SIRI epidemic model:

    ˙S(t)=AμS(t)βS(t)I(t),˙I(t)=βS(t)I(t)(μ+γ)I(t)+δR(t),˙R(t)=γI(t)(μ+δ)R(t), (1.1)

    where S(t),I(t) and R(t) represent the number of susceptible individuals, infectious individuals and recovered individuals at time t, respectively. Assumptions made in the system (1.1) are homogeneous mixing and constant population size. The parameter A is the constant birth rate, μ is the natural death rate, β is the contact rate, and γ is the recovery rate from the infective class. It is assumed that an individual in the recovered class can revert to the infective class with a constant rate δ. Here δ>0 implies that the recovered individuals would lose the immunity, and δ=0 implies that the recovered individuals acquire permanent immunity.

    We note that in system (1.1), the total population size is assumed to be constant. In reality, demographic features which allow the population size to vary should be incorporated into epidemiological models in some cases. For a relatively long-lasting disease or a disease with high death rate, the assumption of logistic growth input of the susceptible seems more reasonable [9]. In [10], Gao and Hethcote investigated a SIRS model with the standard incidence rate, and considered a demographic structure with density-dependent restricted population growth by the logistic equation. In [11], Li et al. studied a SIR epidemic model with logistic growth and saturated treatment, and the existence of the stable limit cycles was obtained. Rencently, there are growing interests in epidemiological models with demographic structures of logistic type [12,13,14,15,16].

    It is worth noting that in the above models, the transmission coefficient is assumed to be constant, and the infected person has the same infectivity during their periodic infection. However, laboratory studies suggest that the infectivity of infectious individuals be different at the differential age of infection [17,18]. Further, it was reported in [19,20] that, age-structure of a population is an important factor which affects the dynamics of disease transmission. In [21], Magal et al. discussed an infection-age model of disease transmission, where both the infectiousness and the removal rate depend on the infection age. In [22], an age structured SIRS epidemic model with age of recovery is studied, and the existence of a local Hopf bifurcation is proved under certain conditions. In [23], Chen et al. considered an SIR epidemic model with infection age and saturated incidence, and established a threshold dynamics by applying the fluctuation lemma and Lyapunov functional.

    Motivated by the works of Chen et al. [23], Gao and Hethcote[10], and Tudor [8], in the present paper, we are concerned with the effects of logistic growth and age of infection on the transmission dynamics of infectious diseases. To this end, we consider the following differential equation system:

    ˙S(t)=rS(t)(1S(t)K)S(t)0β(a)i(a,t)da,i(a,t)t+i(a,t)a=(μ+γ)i(a,t),˙R(t)=γ0i(a,t)da(μ+δ)R(t), (1.2)

    with the boundary condition

    i(0,t)=S(t)0β(a)i(a,t)da+δR(t), (1.3)

    and the initial condition

    X0:=(S(0),i(,0),R(0))=(S0,i0(),R0)X, (1.4)

    where X=R+×L1+(0,)×R+, L1+(0,) is the set of all integrable functions from (0,) into R+=[0,). In system (1.2), S(t) represents the number of susceptible individuals at time t, i(a,t) represents the density of infected individuals with infection age a at time t, and R(t) is the number of individuals who have been infected and temporarily recovered at time t. All parameters in system (1.2) are positive constants, and their definitions are listed in Table 1.

    Table 1.  The definitions of the parameters in system (1.2).
    Parameter Description
    Λ The constant recruitment rate for susceptible populations
    r=Λμ The intrinsic growth rate of susceptible populations
    K The carrying capacity of susceptible population
    μ The rate of natural death
    γ The recovery rate of the infective individuals
    δ The rate at which recovered individuals lose immunity and return to the infective class
    a Age of infection, i.e., the time that has lapsed since the individual became infected
    β(a) Transmission coefficient of the infected individuals at age of infection a

     | Show Table
    DownLoad: CSV

    In the sequel, we further make the following assumption:

    Assumption 1.1 β(a)L+((0,+),R), moreover

    β(a):={β,aτ,0,a(0,τ).

    For convenience, we assume that

    +0β(a)e(μ+γ)ada=1β=(μ+γ)e(μ+γ)τ,

    where e(μ+γ)a is the probability of infected individual to survive to age a and τ>0, β>0.

    The organization of this paper is as follows. In Section 2, we formulate system (1.2) as an abstract non-densely defined Cauchy problem. In Section 3, we study the existence of feasible equilibria of system (1.2). In Section 4, the linearized equation and the characteristic equation of system (1.2) at the interior equilibrium are investigated, respectively. In Section 5, by analyzing corresponding characteristic equation, we discuss the existence of Hopf bifurcation. In Section 6, numerical examples are carried out to illustrate the theoretical results, and sensitivity analysis on several important parameters is carried out.

    In this section, we formulate system (1.2) as an abstract non-densely defined Cauchy problem.

    Firstly, we normalize τ in (1.2) by the timescaling and age-scaling

    ˆa=aτ,ˆt=tτ

    and consider the following distribution

    ˆS(ˆt)=S(τˆt),ˆR(ˆt)=R(τˆt),ˆi(ˆa,ˆt)=τi(τˆt,τˆi).

    Dropping the hat notation, system (1.2) becomes

    {˙S(t)=τ[(Λμ)S(t)(1S(t)K)S(t)0β(a)i(a,t)da],i(a,t)t+i(a,t)a=τ(μ+γ)i(a,t),˙R(t)=τ[γ0i(a,t)da(μ+δ)R(t)], (2.1)

    with the boundary condition i(0,t)=τ[S(t)0β(a)i(a,t)da+δR(t)], and the initial condition S(0)=S00,i(0,)=i0(a)L1((0,+),R),R(0)=R00, where the new function β(a) is defined by

    β(a):={β,a1,0,otherwise,

    and

    +τβe(μ+γ)ada=1β=(μ+γ)e(μ+γ)τ,

    here τ0,β>0.

    Define

    U(t):=+0u(a,t)da,

    where

    U(t)=(S(t)R(t))andu(t,a)=(u1(a,t)u2(a,t)),

    then the first and the third equations of system (2.1) can be rewritten as follows

    {u(a,t)t+u(a,t)a=τCu(a,t),u(0,t)=τG(u1(a,t),u2(a,t)),u(a,0)=u0(a)L1((0,+),R2), (2.2)

    where

    C=(μ00μ+δ),

    and

    G(u1(a,t),u2(a,t))=(G1(u1(a,t),u2(a,t))γ+0i(a,t)da),

    here

    G1(u1(a,t),u2(a,t))=Λ+0u1(a,t)da(1+0u1(a,t)daK)+μ(+0u1(a,t)da)2K+0u1(a,t)da+0β(a)i(a,t)da.

    Let

    w(a,t)=(u(a,t)i(a,t)).

    Accordingly, system (2.1) is equivalent to the following system:

    {w(a,t)t+w(a,t)a=τQw(a,t),w(0,t)=τB(w(a,t)),w(a,0)=w0L1+((0,+),R3), (2.3)

    where

    Q=(μ000μ+δ000μ+γ),

    and

    B(w(a,t))=(G(u1(a,t),u2(a,t))+0u1(a,t)da+0β(a)i(a,t)da+δ+0u2(a,t)da).

    We now consider the following Banach space

    X=R3×L1((0,+),R3)

    with the norm

    (αφ)=∥αR3+φL1((0,+),R3).

    Define the linear operator Lτ:D(Lτ)X by

    Lτ(0R3φ)=(φ(0)φτQφ),

    with D(Lτ)={0R3}×W1,1((0,+),R3)X, and the operator F:¯D(Lτ)X by

    F((0R3φ))=(B(φ)0L1((0,+),R3)).

    Therefore, the linear operator Lτ is non-densely defined due to

    X0:=¯D(Lτ)={0R3}×L1((0,+),R3)X.

    Letting x(t)=(0R3w(,t)), system (2.3) is transformed into the following non-densely defined abstract Cauchy problem

    {dx(t)dt=Lτx(t)+τF(x(t)),t0,x(0)=(0R3w0)¯D(Lτ), (2.4)

    Based on the results in [24] and [25], the global existence, uniqueness and positivity of solutions of system (2.4) are obtained.

    In this section, we study the existence of feasible equilibria of system (2.4).

    Define the threshold parameter R0 by

    R0=K(μ+δ)(μ+γ)μ(μ+γ+δ).

    Suppose that ˉx(a)=(0R3ˉw(a))X0 is a equilibrium of system (2.4). Then we have

    Lτ(0R3ˉw(a))+τF((0R3ˉw(a)))=0,(0R3ˉw(a))¯D(Lτ),

    which is equivalent to

    {ˉw(0)+τB(ˉw(a))=0,ˉw(a)τQˉw(a)=0.

    By direct calculation, we obtain

    ˉw(a)=(ˉu1(a)ˉu2(a)ˉi(a))=(τ[ΛˉS(1ˉSK)+μˉS2KˉS+0β(a)ˉi(a)da]eτμaτγ+0ˉi(a)daeτ(μ+δ)aτ[ˉS+0β(a)ˉi(a)da+δˉR]eτ(μ+γ)a), (3.1)

    where ˉS=+0u1(a,t)da,ˉR=+0u2(a,t)da. From (3.1), it is easy to show that

    ˉi(a)=τ[ˉS+0β(a)ˉi(a)da+δˉR]eτ(μ+γ)a. (3.2)

    Integrating Eq (3.2), we get

    +0β(a)ˉi(a)da=ˉS+0β(a)ˉi(a)da+δˉR (3.3)

    and

    +0ˉi(a)da=1μ+γ+0β(a)ˉi(a)da. (3.4)

    We derive from the first and second equations of Eq (3.1) that

    rˉS(1ˉSK)ˉS+0β(a)ˉi(a)da=0,ˉR=γμ+δ+0ˉi(a)da. (3.5)

    This, together with (3.4), yields

    ˉR=γμ+δ1μ+γ+0β(a)ˉi(a)da. (3.6)

    On substituting (3.6) into (3.3), we obtain that

    ˉS=μ(μ+γ+δ)(μ+δ)(μ+γ). (3.7)

    It follows from (3.5) that

    +0β(a)ˉi(a)da=rμ(μ+γ+δ)K(μ+δ)(μ+γ)(R01). (3.8)

    We therefore follows from (3.6) and (3.8) that

    ˉR=rγμ(μ+γ+δ)K(μ+δ)2(μ+γ)2(R01). (3.9)

    On substituting (3.7)–(3.9) into (3.2), we get

    ˉi(a)=τrμ(μ+γ+δ)K(μ+δ)(μ+γ)(R01)eτ(μ+γ)a.

    Based on the discussions above, we have the following result.

    Lemma 3.1. System (2.4) always has the equilibrium

    ˉx0(a)=(0R3(τμKeτμa0L1((0,),R)0L1((0,),R))).

    In addition, if R0>1, there exists a unique positive equilibrium

    ˉx(a)=(0R3ˉw(a))=(0R3(τμ2(μ+γ+δ)(μ+δ)(μ+γ)eτμaτrγμ(μ+γ+δ)K(μ+δ)(μ+γ)2(R01)eτ(μ+δ)aτrμ(μ+γ+δ)K(μ+δ)(μ+γ)(R01)eτ(μ+γ)a)).

    Correspondingly, for system (1.2), we have the following result.

    Theorem 3.1. System (1.2) always has a disease-free steady state E0(K,0,0). If R0>1, system (1.2) has a unique endemic steady state E(S,i(a),R), where

    S=μ(μ+γ+δ)(μ+δ)(μ+γ),i(a)=τrμ(μ+γ+δ)K(μ+δ)(μ+γ)(R01)eτ(μ+γ)a,R=rγμ(μ+γ+δ)K(μ+δ)2(μ+γ)2(R01).

    In this section, we investivate the linearized equation of (2.4) around the positive equilibrium ˉx(a), and the characteristic equation of (2.4) at ˉx(a), respectively.

    Making the change of variable y(t):=x(t)ˉx(a), system (2.4) becomes

    {dy(t)dt=Lτy(t)+τF(y(t)+ˉx(a))τF(ˉx(a)),t0,y(0)=(0R3w0ˉw(a)):=y0¯D(Lτ). (4.1)

    Accordingly, the linearized equation of system (4.1) around the origin is

    dy(t)dt=Lτy(t)+τDF(ˉx(a))y(t),t0,y(t)X0,

    where

    τDF(ˉx(a))(0R3φ)=(τDB(ˉw(a))(φ)0L1((0,+),R3)),(0R3φ)D(Lτ),

    with

    DB(ˉw(a))(φ)=(Λ2rˉSK+0β(a)ˉi(a)da0000γ+0β(a)ˉi(a)daδ0)×+0φ(a)da+(00ˉS00000ˉS)×+0β(a)φ(a)da.

    After that, system (4.1) can be rewritten as

    dy(t)dt=Lτy(t)+F(y(t)),t0, (4.2)

    where the linear operator L:=Lτ+τDF(ˉx(a)) and

    F(y(t))=τF(y(t)+ˉx(a))τF(ˉx(a))τDF(ˉx(a))y(t)

    satisfying F(0)=0,DF(0)=0.

    In the following, we give the characteristic equation of (2.4) at the positive equilibrium. By means of the method used in [26], we obtain the following lemma.

    Lemma 4.1. Let λΩ={λC:Re(λ)>μτ},λρ(Lτ) and

    (λILτ)1(αψ)=(0R3φ)φ(a)=ea0(λI+τQ)dlα+a0eas(λI+τQ)dlψ(s)ds (4.3)

    with (αψ)X and (0R3φ)D(Lτ), where Lτ is a Hille-Yosida operator and

    (λILτ)n1(Re(λ)+μτ)n,λΩ,n1. (4.4)

    Let L0 be the part of Lτ in ¯D(Lτ), that is L0:=D(L0)XX. For (0R3φ)D(L0), we have

    L0(0R3φ)=(0R3^L0(φ)),

    where ^L0(φ)=φτQφ with D(^L0)={φW1,1((0,+),R3):φ(0)=0}.

    Note that τDF(ˉx):D(Lτ)XX is a compact bounded linear operator. It follows from (4.4) that

    TL0(t)eμτt,t0.

    Therefore

    ω0,ess(L0)ω0(L0)μτ.

    By using the perturbation results of [35], we get

    ω0,ess((Lτ+τDF(ˉx))0)μτ<0.

    Hence, we have the following result.

    Lemma 4.2. The linear operator Lτ is a Hille-Yosida operator, and its part (Lτ)0 in ¯D(Lτ) satisfies

    ω0,ess((Lτ)0)<0.

    Set λΩ. Since λILτ is invertible, it follows that λILτ is invertible if and only if IτDF(ˉx)(λILτ)1 is invertible, and

    (λILτ)1=(λI(Lτ+τDF(ˉx)))1=(λILτ)1(IτDF(ˉx)(λILτ)1)1.

    We now consider

    (IτDF(ˉx)(λILτ)1)(αφ)=(ξψ),

    which yields

    (αφ)τDF(ˉx)(λILτ)1(αφ)=(ξψ).

    It is easy to show that

    {ατDB(ˉw)(ea0(λI+τQ)dlα)=ξ+τDB(ˉw)(a0eas(λI+τQ)dlφ(s)ds),φ=ψ.

    Taking the formula of DB(ˉw) into consideration, we obtain

    {Δ(λ)α=ξ+K(λ,ψ),φ=ψ, (4.5)

    where

    K(λ,ψ)=τDB(ˉw)(a0eas(λI+τQ)dlψ(s)ds), (4.6)

    and

    Δ(λ)=IτDB(ˉw)(ea0(λI+τQ)dlα)=Iτ(Λ2rˉSK+0β(a)ˉi(a)da0000γ+0β(a)ˉi(a)daδ0)+0ea0(λI+τQ)dldaτ(00ˉS00000ˉS)+0β(a)ea0(λI+τQ)dlda. (4.7)

    From (4.5), whenever Δ(λ) is invertible, we have

    α=(Δ(λ))1(ξ+K(λ,ψ)).

    Using a similar argument as in [27], it is easy to verify the following result.

    Lemma 4.3. The following results hold

    (i) σ(Lτ)Ω=σp(Lτ)Ω={λΩ:det(Δ(λ))=0};

    (ii) If λρ(Lτ)Ω, we have the formula for resolvent

    (λILτ)1(αψ)=(0R3φ), (4.8)

    where

    φ(a)=ea0(λI+τQ)dl(Δ(λ))1[ξ+K(λ,ψ)]+a0eas(λI+τQ)dlψ(s)ds,

    with Δ(λ) and K(λ,ψ) given by (4.6) and (4.7).

    Under Assumption 1.1, it therefore follows from (4.7) that

    Δ(λ)=(1(Λ2rˉSK+0β(a)ˉi(a)da)τλ+τμ0ˉSτβe(λ+τ(μ+γ))λ+τ(μ+γ)01γτλ+τ(μ+γ)+0β(a)ˉi(a)daτλ+τμδτλ+τ(μ+δ)1ˉSτβe(λ+τ(μ+γ))λ+τ(μ+γ)) (4.9)

    From (4.6), we obtain the characteristic equation of system (2.4) at the positive equilibrium ˉx(a) as follows:

    det(Δ(λ))=λ3+τp2λ2+τ2p1λ+τ3p0+(τq2λ2+τ2q1λ+τ3q0)eλ(λ+τμ)(λ+τ(μ+γ))(λ+τ(μ+δ))f(λ)g(λ)=0, (4.10)

    where

    f(λ)=λ3+τp2λ2+τ2p1λ+τ3p0+(τq2λ2+τ2q1λ+τ3q0)eλ,g(λ)=(λ+τμ)(λ+τ(μ+γ))(λ+τ(μ+δ)),

    here

    p0=μ(μ+δ+γ)rˉSK,p1=μ(μ+δ+γ)+(2μ+δ+γ)rˉSK,p2=2μ+δ+γ+rˉSK,q0=ˉS(μ+γ)(μ+δ)(r2rˉSK),q1=ˉS(μ+γ)(μ+δ)+ˉS(μ+γ)(r2rˉSK),q2=ˉS(μ+γ).

    Letting λ=τζ, then

    f(λ)=f(τζ)=τ3g(ζ)=τ3[ζ3+p2ζ2+p1ζ+p0+(q2ζ2+q1ζ+q0)eτζ]. (4.11)

    It is easy to show that

    {λΩ:det(Δ(λ))=0}={τζΩ:g(ζ)=0}.

    In this section, by applying Hopf bifurcation theory [26], we are concerned with the existence of Hopf bifurcation for the Cauchy problem (2.4) by regarding τ as the bifurcation parameter.

    From (4.11), we have

    g(ζ)=ζ3+p2ζ2+p1ζ+p0+(q2ζ2+q1ζ+q0)eτζ. (5.1)

    For any τ0, if R0>1, it is easy to show that

    g(0)=p0+q0=μ(μ+δ+γ)(rrˉSK)>0.

    Therefore, ζ=0 is not an eigenvalue of Eq (5.1). Furthermore, when τ=0, Eq (5.1) reduces to

    ζ3+(p2+q2)ζ2+(p1+q1)ζ+p0+q0=0. (5.2)

    A direct calculation shows that

    p2+q2=(μ+δ)2+δγμ+δ+rˉSK>0,

    and

    (p2+q2)(p1+q1)(p0+q0)=rˉSK[((μ+δ)2+γδμ+δ)2+rˉSK((μ+δ)2+γδμ+δ)+ˉS(μ+γ)(rrˉSK)]+μδγ(μ+δ+γ)(μ+δ)2(rrˉSK)>0.

    Hence, by Routh-Hurwitz criterion, when τ=0, we see that the equilibrium ˉx(a) is locally asymptotically stable if R0>1; and ˉx(a) is unstable if R0<1.

    Substituting ζ=iω(ω>0) into Eq (5.1) and separating the real and imaginary parts, one obtains that

    ω3p1ω=(q2ω2q0)sinωτ+q1ωcosωτ,p2ω2p0=(q2ω2q0)cosωτ+q1ωsinωτ. (5.3)

    Squaring and adding the two equations of (5.3), it follows that

    ω6+h2ω4+h1ω2+h0=0, (5.4)

    where

    h0=p20q20,h1=p21q21+2q0q22p0p2,h2=p22q222p1. (5.5)

    Letting z=ω2, Eq (5.4) can be written as

    z3+h2z2+h1z+h0=0. (5.6)

    Denote

    h(z)=z3+h2z2+h1z+h0=0,Δ=h223h1,

    and define

    z1=h2+Δ3,z2=h2Δ3.

    By a similar argument as in [28], we have the following result.

    Lemma 5.1. [28]. For the polynomial equation (5.6), the following results hold true:

    (i) If h0<0, then Eq (5.6) has at least one positive root.

    (ii) If h00 and Δ<0, then Eq (5.6) has no positive root.

    (iii) If h00 and Δ0, then Eq (5.6) has at least one positive root if one of z1>0 and h(z1)0.

    Noting that

    h2=μ2+(rˉSK)2+(μ+δ+γ)2(2μδ+δ2)(μ+δ)2>0,

    without loss of generality, we may assume that Eq (5.6) has two positive roots denoted respectively as z1 and z2. Then Eq (5.4) has two positive roots ωk=zk(k=1,2). Further, from (5.3), we have

    τ(j)k={1ωk{arccos((q1p2q2)ω4k+(p2q0+p0q2p1q1)ω2kp0q0q21ω2k+(q2ω2kq0)2)+2jπ},Θ0, 1ωk{2πarccos((q1p2q2)ω4k+(p2q0+p0q2p1q1)ω2kp0q0q21ω2k+(q2ω2kq0)2)+2jπ},Θ<0, (5.7)

    where k=1,2;j=0,1,, and

    Θ=q2ω5k+(p2q1q0p1q2)ω3k+(p1q0p0q1)ωkq21ω2k+(q2ω2kq0)2.

    Based on the above discussion, we have the following result.

    Theorem 5.1. Let Assumption 1.1 and R0>1 hold. If ωk is a positive root of Eq (5.4) and q10, then

    dg(ζ)dζ|ζ=iωk0.

    Therefore ζ=iωk is a simple root of Eq (5.1).

    Proof. It follows from (5.1) that

    dg(ζ)dζ|ζ=iωk=3ω2k+i2p2ωk+p1+(i2q2ωk+q1τ(j)k(q2ω2k+iq1ωk+q0))eiωkτ(j)k

    and

    [3ζ2+2p2ζ+p1+(2q2ζ+q1τ(q2ζ2+q1ζ+q0))eτζ]dζ(τ)dτ=ζ(q2ζ2+q1ζ+q0)eτζ.

    Suppose that dg(ζ)dζ|ζ=iωk=0, then

    iωk(q2ω2k+iq1ωk+q0)eiωkτ(j)k=0. (5.8)

    Separating the real and imaginary parts in Eq (5.8), we obtain

    (q0ωkq2ω3k)sinωkτ(j)kq1ω2kcosωkτ(j)k=0,(q0ωkq2ω3k)cosωkτ(j)k+q1ω2ksinωkτ(j)k=0. (5.9)

    Squaring and adding the two equations of (5.9), we derive that

    (q0ωkq2ω3k)2+(q1ω2k)2=0,

    which mean that

    q0ωkq2ω3k=0andq1ω2k=0.

    Since ωk>0, it follows that

    q0ωkq2ω3k=0andq1=0,

    which leads to a contradiction. Hence, we have

    dg(ζ)dζ|ζ=iωk0.

    Let ζ(τ)=α(τ)+iω(τ) be a root of Eq (5.1) satisfying α(τ(j)k)=0,ω(τ(j)k)=ωk, where

    τ0=mink{1,2}{τ(0)k},j=0,1,2,,ω0=ωk0. (5.10)

    Lemma 5.2. Let Assumption 1.1 and R0>1 hold. If zk=ω2k,h(zk)0 and q10, then

    Re[dζ(τ)dτ|τ=τ(j)k]0,

    and dReζ(τ)/dτ and h(zk) have the same sign.

    Proof. Differentiating the two sides of Eq (5.1) with respect to τ, we get

    (dζdτ)1=3ζ2+2p2ζ+p1ζ(ζ3+p2ζ2+p1ζ+p0)+2q2ζ+q1ζ(q2ζ2+q1ζ+q0)τζ. (5.11)

    On substituting ζ=iωk into Eq (5.1), by calculating, we have

    Re[(d(Reζ)dτ)1|ζ=iωk]=3ω4k+2(p222p1)ω2k+p212p0p2(ω2+p1)2ω2k+(p0p2ω2k)22q22ω2k+q212q0q2q21ω2k+(q0q2ω2k)2.

    A direct calculation shows that

    sign{d(Reζ)dτ}ζ=iωk=sign{Re(dζdτ)1}ζ=iωk=sign[3ω4k+2(p222p1)ω2k+p212p0p2(ω2kp1)2ω2k+(p0p2ω2k)2+2q22ω2kq21+2q2q0q21ω2k+(q0q2ω2k)2].

    From Eq (5.4), we get

    (ω2kp1)2ω2k+(p0p2ω2k)2=q21ω2k+(q0q2ω2k)2.

    It therefore follows that

    sign{d(Reζ)dτ}ζ=iωk=sign[h(zk)(q3ω2kq1)2ω2k+(q0q2ω2k)2].

    Since zk>0, we conclude that Re[dζ(τ)/dτ] and h(zk) have the same sign.

    Noting that when τ=0, the equilibrium x(a) of (2.4) is locally asymptotically stable if R0>1, from what has been discussed above, we have the following results.

    Theorem 5.2. Let τ(j)k and ω0,τ0 be defined by (5.7) and (5.10), respectively. If Assumption 1.1 and R0>1, q10 hold.

    (i) the endemic steady state E of system (1.2) is locally asymptotically stable for all τ0 if h00 and Δ0.

    (ii) the endemic steady state E is asymptotically stable for τ[0,τ0) if h0<0 or h00,Δ>0,z1>0 and h(z1)0.

    (iii) system (1.2) undergoes a Hopf bifurcation at endemic steady state E when τ=τ(j)k (j=0,1,2,) if the conditions as stated in (ii) are satisfied and h(zk)0.

    In this section, numerical simulations will be given to illustrate the theoretical results in Section 5. Further, sensitivity analysis is used to quantify the range of variability in threshold parameter and to identify the key factors giving rise to threshold parameter, which is helpful to design treatment strategies.

    In this section, we give a numerical example to illustrate the main results in Section 5.

    Based on the research works of tuberculosis [29,30,32], parameter values of system (1.2) are summarized in Table 2, and the maximum infection age is 60. Denote the numbers of infected individuals at time t as I(t)=1000i(t,a)da, and

    β(a):={1135e1135τ,aτ,0,a(0,τ).
    Table 2.  Parameter values for the age-structured SIRI epidemic model (1.2).
    Parameter Symbol value Source
    The carrying capacity of susceptible population K(human) 5 Assumed
    The intrinsic growth rate of susceptible populations r(peryear) 0.1 Assumed
    The rate of natural death μ(peryear) 1/70 [29,30]
    The rate at which recovered individuals lose immunity and return to the infective class δ(peryear) 0.01 Assumed
    The recovery rate of the infective individuals γ(peryear) 0.3 [31]

     | Show Table
    DownLoad: CSV

    By calculation, we have R0=8.2379>1,h0=1.1989×107<0 and h(zk)=2.7846×104>0. In this case, system (1.2) has an endemic steady state E(0.6070,0.0879τe0.3143τa,3.4534). By calculation, we obtain ω0=0.0565 and τ0=14.3556. By Theorem 5.1, we see that the endemic steady state E is locally asymptotically stable if τ[0,τ0) and is unstable if τ>τ0. Further, system (1.2) undergoes a Hopf bifurcation at E when τ=τ0. Numerical simulation illustrates the result above (see, Figures 1 and 2).

    Figure 1.  Numerical solutions of system (1.2) with τ=8<τ0=14.3556. (a) the trajectories of susceptible individuals S(t); (b) the trajectories of infected individuals I(t); (c) the trajectories of recovered individuals R(t); (d) the dynamical behavior of infected individuals i(a,t).
    Figure 2.  Numerical solutions of system (1.2) with τ=16>τ0=14.3556. (a) the trajectories of susceptible individuals S(t); (b) the trajectories of infected individuals I(t); (c) the trajectories of recovered individuals R(t); (d) the dynamical behavior of infected individuals i(a,t).

    Remark. From Figure 1, we see that when the bifurcation parameter τ is less than the critical value τ0, the endemic steady state E of system (1.2) is locally asymptotically stable. From Figure 2, we observe that E losses its stability and Hopf bifurcation occurs when τ crosses τ0 to the right (τ>τ0). This implies that the age, i.e., infection period τ is the key factor that causes the endemic steady state E to become unstable and the appearance of Hopf bifurcation.

    Sensitivity analysis is used to quantify the range of variables in threshold parameter and identify the key factors giving rise to threshold parameter. In [33,34], Latin hypercube sampling (LHS) is found to be a more efficient statistical sampling technique which has been introduced to the field of disease modelling. LHS allows an un-biased estimate of the threshold parameter, with the advantage that it requires fewer samples than simple random sampling to achieve the same accuracy.

    By analysis of the sample derived from Latin hypercube sampling, we can obtain large efficient data in respect to different parameters of R0. Figure 3 shows the scatter plots of R0 in respect to K, μ, δ and γ, which implies that δ is a positively correlative variable with R0, while μ is a negatively correlative variable. But the correlation between K, γ and R0 is not clear. In [34], Marino et al. mentioned that Partial Rank Correlation Coefficients (PRCCs) provide a measure of the strength of a linear association between the parameters and the threshold parameter. Furthermore, PRCCs are useful for identifying the most important parameters. The positive or negative of PRCCs respectively denote the positive or negative correlation with the threshold parameter, and the sizes of PRCCs measure the strength of the correlation. As can been seen in Figure 4, K, δ and γ are positively correlative variables with R0 while μ is negatively correlative variables.

    Figure 3.  Scatter plots of R0 in respect to K, μ, δ and γ.
    Figure 4.  Tornado plot of PRCCs in regard to R0.

    By selecting different parameter values, we can explore the influence of the parameters μ and δ on the numbers of infected individuals at time t, which is denoted as I(t). As shown in Figure 5, increasing the natural death rate μ and decreasing the rate at which recovered individuals return to the infective class will have a positive impact on I(t) to some extent, which means that the influence of μ and δ on I(t) is consistent with that on R0.

    Figure 5.  The influence of μ and δ on the numbers of infected individuals at time t, where τ=2 and the values of other parameters are consistent with those in Figure 1. (a) the trajectories of I(t) corresponding to different values of μ; (b) the trajectories of I(t) corresponding to different values of δ.

    This work was supported by the National Natural Science Foundation of China (Nos. 11871316, 11801340), the Natural Science Foundation of Shanxi Province (Nos. 201801D121006, 201801D221007).

    The authors declare that they have no competing interests.



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